# 线性代数网课代修|数值线性代数代写Numerical linear algebra代考|MATH307

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• 数值分析
• 高等线性代数
• 矩阵论
• 优化理论
• 线性规划
• 逼近论

## 线性代数作业代写linear algebra代考|Basic Properties

A linear system has a unique solution, infinitely many solutions, or no solution. To discuss this we first consider the real case, and a homogeneous underdetermined system.

Lemma 1.3 (Underdetermined System) Suppose $A \in \mathbb{R}^{m \times n}$ with $m<n$. Then there is a nonzero $x \in \mathbb{R}^{n}$ such that $\boldsymbol{A x}=\mathbf{0}$.

Proof Suppose $\boldsymbol{A} \in \mathbb{R}^{m \times n}$ with $m<n$. The $n$ columns of $\boldsymbol{A}$ span a subspace of $\mathbb{R}^{m}$. Since $\mathbb{R}^{m}$ has dimension $m$ the dimension of this subspace is at most $m$. By Lemma $1.1$ the columns of $\boldsymbol{A}$ must be linearly dependent. It follows that there is a nonzero $\boldsymbol{x} \in \mathbb{R}^{n}$ such that $\boldsymbol{A} \boldsymbol{x}=\mathbf{0}$.
A square matrix is either nonsingular or singular.
Definition 1.7 (Real Nonsingular or Singular Matrix) A square matrix $A \in$ $\mathbb{R}^{n \times n}$ is said to be nonsingular if the only real solution of the homogeneous system $\boldsymbol{A x}=\mathbf{0}$ is $\boldsymbol{x}=\mathbf{0}$. The matrix is singular if there is a nonzero $\boldsymbol{x} \in \mathbb{R}^{n}$ such that $A \boldsymbol{x}=0$.

Theorem 1.6 (Linear Systems; Existence and Uniqueness) Suppose $A \in \mathbb{R}^{n \times n}$. The linear system $\boldsymbol{A} \boldsymbol{x}=\boldsymbol{b}$ has a unique solution $\boldsymbol{x} \in \mathbb{R}^{n}$ for any $\boldsymbol{b} \in \mathbb{R}^{n}$ if and only if the matrix $\boldsymbol{A}$ is nonsingular.

Proof Suppose $\boldsymbol{A}$ is nonsingular. We define $\boldsymbol{B}=[\boldsymbol{A} \boldsymbol{b}] \in \mathbb{R}^{n \times(n+1)}$ by adding a column to $\boldsymbol{A}$. By Lemma $1.3$ there is a nonzero $z \in \mathbb{R}^{n+1}$ such that $\boldsymbol{B} z=\mathbf{0}$. If we write $z=\left[\begin{array}{c}\tilde{z} \ z_{n+1}\end{array}\right]$ where $\tilde{z}=\left[z_{1}, \ldots, z_{n}\right]^{T} \in \mathbb{R}^{n}$ and $z_{n+1} \in \mathbb{R}$, then
$$\boldsymbol{B} z=[\boldsymbol{A} \boldsymbol{b}]\left[\begin{array}{c} \tilde{z} \ z_{n+1} \end{array}\right]=\boldsymbol{A} \tilde{z}+z_{n+1} \boldsymbol{b}=\mathbf{0} .$$

## 线性代数作业代写linear algebra代考|The Inverse Matrix

Suppose $\boldsymbol{A} \in \mathbb{C}^{n \times n}$ is a square matrix. A matrix $\boldsymbol{B} \in \mathbb{C}^{n \times n}$ is called a right inverse of $\boldsymbol{A}$ if $\boldsymbol{A B}=\boldsymbol{I}$. A matrix $\boldsymbol{C} \in \mathbb{C}^{n \times n}$ is said to be a left inverse of $\boldsymbol{A}$ if $\boldsymbol{C} \boldsymbol{A}$. We say that $\boldsymbol{A}$ is invertible if it has both a left- and a right inverse. If $\boldsymbol{A}$ has a right inverse $B$ and a left inverse $\boldsymbol{C}$ then
$$\boldsymbol{C}=\boldsymbol{C I}=\boldsymbol{C}(\boldsymbol{A B})=(\boldsymbol{C} \boldsymbol{A}) \boldsymbol{B}=\boldsymbol{I} \boldsymbol{B}=\boldsymbol{B}$$
and this common inverse is called the inverse of $A$ and denoted by $A^{-1}$. Thus the inverse satisfies $\boldsymbol{A}^{-1} \boldsymbol{A}=\boldsymbol{A} \boldsymbol{A}^{-1}=\boldsymbol{I}$.

We want to characterize the class of invertible matrices and start with a lemma.
Theorem $1.8$ (Product of Nonsingular Matrices) If $\boldsymbol{A}, \boldsymbol{B}, \boldsymbol{C} \in \mathbb{C}^{n \times n}$ with $\boldsymbol{A} \boldsymbol{B}=$ $\boldsymbol{C}$ then $\boldsymbol{C}$ is nonsingular if and only if both $\boldsymbol{A}$ and $\boldsymbol{B}$ are nonsingular. In particular, if either $\boldsymbol{A} \boldsymbol{B}=\boldsymbol{I}$ or $\boldsymbol{B} \boldsymbol{A}=\boldsymbol{I}$ then $\boldsymbol{A}$ is nonsingular and $\boldsymbol{A}^{-1}=\boldsymbol{B}$.

Proof Suppose both $\boldsymbol{A}$ and $\boldsymbol{B}$ are nonsingular and let $\boldsymbol{C} \boldsymbol{x}=\mathbf{0}$. Then $\boldsymbol{A B}=\mathbf{0}$ and since $\boldsymbol{A}$ is nonsingular we see that $\boldsymbol{B} \boldsymbol{x}=\mathbf{0}$. Since $\boldsymbol{B}$ is nonsingular we have $\boldsymbol{x}=\mathbf{0}$. We conclude that $\boldsymbol{C}$ is nonsingular.

## 线性代数作业代写linear algebra代考|Basic Properties

$$\boldsymbol{B} z=[\boldsymbol{A} \boldsymbol{b}]\left[\tilde{z} z_{n+1}\right]=\boldsymbol{A} \tilde{z}+z_{n+1} \boldsymbol{b}=\mathbf{0} .$$

## 线性代数作业代写linear algebra代考|The Inverse Matrix

$$\boldsymbol{C}=\boldsymbol{C I}=\boldsymbol{C}(\boldsymbol{A B})=(\boldsymbol{C A}) \boldsymbol{B}=\boldsymbol{I} \boldsymbol{B}=\boldsymbol{B}$$

# 计量经济学代写

## 在这种情况下，如何学好线性代数？如何保证线性代数能获得高分呢？

1.1 mark on book

【重点的误解】划重点不是书上粗体，更不是每个定义，线代概念这么多，很多朋友强迫症似的把每个定义整整齐齐用荧光笔标出来，然后整本书都是重点，那期末怎么复习呀。我认为需要标出的重点为

A. 不懂，或是生涩，或是不熟悉的部分。这点很重要，有的定义浅显，但证明方法很奇怪。我会将晦涩的定义，证明方法标出。在看书时，所有例题将答案遮住，自己做，卡住了就说明不熟悉这个例题的方法，也标出。

B. 老师课上总结或强调的部分。这个没啥好讲的，跟着老师走就对了

C. 你自己做题过程中，发现模糊的知识点

1.2 take note

1.3 understand the relation between definitions